Differential Equations

By Bobby Neal Winters

These days I spent a lot of time thinking about teaching, about learning, about setting up systems wherein the first will facilitate the second.  We want to teach our students certain skills and certain content, but there are other things, things more mysterious that we want to happen too.  One of these things is called knowledge integration.

When I was in college in the early 1980s, there was a group of us who were being educated in the sciences.  This was on the down-side of the wave that was caused by Sputnik and the Cold War. I still think of that era as the good old days.  Sure we were worried about nuclear annihilation, but we were working.

There was a group of use who were all taking the same classes.  We would go from computer programming to calculus and from calculus to physics.  Occasionally there were those in the group who were more experienced and worldly who would give the rest of us the low-down on how the world worked.  It was much like learning about sex.  There was the official story that the grown-ups gave us, about doing things responsibly and preparing for the future, that didn’t sound all that exciting at all, but there is the unofficial story from our near-peers which is grittier and somehow more attractive as it is full of all sorts of shortcuts and inside information.

For one thing, if you were in engineering you wanted to be in civil engineering if the democrats were in, aeronautical engineering if the republicans were, and electrical engineering if you weren’t sure, but that electrical engineering was hard. They would also talk about the hard classes, the ones that you should put off as soon as possible: COBOL, Organic Chemistry, and Differential Equations.

These classes were so hard that you begin to hear about them from your college-age friends while you were still in high school.  They were the Unholy Trinity of the Sciences.  Of these, I took Differential Equations and I took it the first semester of my Sophomore year.  I’d had every intention of putting it off, honest, but Ken Brady, who was the Acting Chair of the Department of Mathematics when I started college wouldn’t hear of that.  I needed to get it in as early as possible so I could go on to take more challenging courses.

I will grant my near-peers one thing.  Differential Equations was one of the most challenging courses I’d had in my life up to that point in spite of having been very well prepared for it.  Its one and only thing in common with sex is that you can never understand the experience until you’ve been through it.  Indeed, this is even more so in the case of Differential Equations as nature has prepared us for sex in a way that it hasn’t Differential Equations. This having been said, let me try to explain it in non-technical terms.

You start mathematics with algebra. Then you take trigonometry which uses algebra.  Then you go into Calculus which in those days was divided into Calculus I and Calculus II.  In calculus, you do use some algebra and some trigonometry.  There will be sections here and there where you as a student are required to recall some algebraic trivia or some arcane formulas from trigonometry, but those instances are fairly well quarantined from each other. There is breathing room around them.  There is time to sit back and say, “Yep, that was kind of hard, but I lived through it.”

Differential Equations is different.  To begin with, there is the tacit assumption on part of the teacher that you remember with perfect precision every mathematical activity you’ve ever participated in in your entire life. You know how to solve every polynomial equations; you know how to evaluate every obscure integral; you are comfortable, nay, accomplished with the arithmetic of complex numbers.

I’ve since had the opportunity to teach this course, and I stand amazed at the amount of work that my teacher, Mr. Phillip Briggs, was able to get out of us.  The man didn’t have a doctorate, but that didn’t matter.  He had the knack of getting us to work.  You may remember the character Fezziwig from Charles Dickens’ A Christmas Carol.  Scrooge tells the Ghost of Christmas Past, “He has the power to make us happy or unhappy.”  Well, Mr. Briggs had the power to make us work our backsides off.

In Differential Equations, you are learning some new concepts, but those new concepts require that you remember some old ones.  In algebra, we learn about solving polynomial equations and obtaining their roots.  We also learn about exponential functions.  In Differential Equations there is a technique where in you use both of those things, plus keep track of some completely new and arcane rules at the same time.  Then they throw complex numbers into the mix just for good measure.

Then there is the amount of work involved.  In algebra, most problems can be solved in a few lines.  In calculus, most can be completed in half a page.  In Differential Equations, especially when you start using infinite series, the solution of one problem can literally go on for several pages, and on any line of those several pages, your solution might easily go awry. The text we used had the answer to every problem, so that when you were done with your several pages of work you could check to see in you were right.  If you weren’t--which did happen with astonishing frequency--you had to start all over.  Which I did, even though--and this was part of Mr. Briggs’ genius--the teacher never took up homework!

I knew there was something special happening at the time, but I didn’t know the name for it and didn’t learn for many years later.  I was integrating my knowledge.  We all were.  We were taking things that we had learned in separate, isolated settings and bringing them together in a new setting.  In applying our algebra in a new setting, we were making it a part of a larger world.

To be fair, this was happening in lesser degrees in other courses like physics where we applied math to physical problems, but that didn’t use such a broad variety of mathematics and didn’t use it so intensely.

Differential Equations served as a crucible for the Knowledge Integration, but the work that Mr. Briggs got out of use was the sine qua non.  Like Jewel said, “There ain’t nothing for free.”

Trigonometry

By Bobby Neal Winters

Hurt so good

Come on baby, make it hurt so good

Sometimes love don't feel like it should

You make it hurt so good

--John Mellancamp

Before the seventh or eighth grade I would not have said I was good at math.  I was good at science.  I had a great memory. I once destroyed an encyclopedia salesman who came by our rural home when I was ten or twelve years old.  He pulled out his product, opened it to a page that featured a picture of a skeleton, and began is his pitch to my mom.

“This can help your son with is science home work,” he said.  “Every part of the body has a scientific name.  They can’t just call a shoulder blade a shoulder blade.”

“Scapula,” I said.

“What?”

“It’s called a scapula.”

He looked a little thrown off.

“Or a breast bone a breast bone,” he continued.

“Sternum,” I said.

“Or the arm bones,” he proffered.

“Radius, ulna, and humerus.”

He went down and Momma smiled.

But that is just memory.  I loathed mathematics as I knew it.  Arithmetic was my enemy.  Multiplication was hard.  Long division was almost impossible.  By crying, I manipulated Momma into doing it for me.  There was--and is--some block in my head that keeps me from doing it.  Professionals have told me that I have dyslexia, but I’ve never been tested.

The tide in the war between mathematics and me began to turn in the seventh grade when the teachers began to introduce elements of algebra into class. When I took algebra in the ninth grade, I didn’t consider it a chore any more.  In my Sophomore year, I took geometry, and it was as if the scales fell from my eyes.  It was mathematics without arithmetic.  

I thought I was in heaven.

The geometry class had seniors in it, and I cleaned their collective clocks. This made mathematics very important to my self-worth.  I wanted more of it.

My chance came the next year when Algebra II and Trigonometry were both offered. This was a problem because Trigonometry requires the skill set taught in Algebra II, which at that time included the algebra of rational expressions and quadratic equations both of which are needed in Trigonometry.

My teacher, Mr. Sloan, told me that I was doing things out of order but they would let me.  Trigonometry was only offered every other year because I went to a small country high school that simply didn’t have the staff to offer it more frequently.  If I was going to get my trigonometry in before I went to college, I’d just have to do it this way.

For these reasons, Trigonometry became a crucible for my mathematical education.  It was hard because you do need Algebra II in order to do Trigonometry.  There were times when I’d cry while doing my trig homework, but this time Momma couldn’t do it for me because she’d never had it. I had to do it myself.

Two things helped. One of these was that trigonometry has a high content of geometry.  The confidence I’d developed in geometry carried over.  The other was that I was committed to this.  My self-image and ego were on the line.  I did my Algebra II homework first to get it out of the way, and then I did my Trigonometry homework twice.  

This is something that I don’t often share with students but maybe I should.  Doing your homework is good, but doing it twice is better.  It might even be more than twice as good.  You repeat the skill and reinforce it, but you know where it is going and you do it with more confidence.

Writing an assignment the second time is something I’d avoided before then even though it had been suggested to me on multiple occasions. My handwriting is terrible.  I print almost everything and even that is terrible. This is one of the reasons I am suspected of having dyslexia.  So the reason all of my teachers wanted me to recopy was the very reason I wouldn’t.  It was hard.

This time my ego was so tied-up in the subject I finally took the advice.  Aesthetically speaking, the results weren’t good on the second draft, but they were better than the first.

(As an aside, my teachers had always told me to just take my time with my handwriting.  While there is a lot of virtue to that, the subsequent years have proven to me more was needed than just that.  I’ve made new copies of my lecture notes from year-to-year, slowly recopying everything. The results are legible, but barely, and I certainly never have achieved a “good hand.”  There are limits.)

So my course in trigonometry was a struggle for me.  It was an example of what is called productive pain. Okay, what is it and what’s it good for?

Trigonometry is the study of triangles.  Triangles are geometric objects, but in trigonometry we use numbers and algebra to study them.  There are two major aspect to the course: practical and theoretical.  

The practical part consists of learning various techniques, including  the Law of Sines and the Law of Cosines, in order to measure the sides and the angles of a triangle from known information.  There are certain situations where you can get back a whole lot more information than you put in.  Students, especially those who are of a practical turn of mind, seem to appreciate this part of the course as it can be immediately applied.

They are not so sanguine about the theoretical part of the course.  Those who’ve had the course will know that I am referring to the various identities one is force to learn, manipulate, and prove to be true.  Students don’t like trigonometric identities.  Indeed, hate is not too strong a word to use here.

The proofs that we make students perform in these identities are far from intuitive.  They are like mazes in that students can make a wrong turn and have a hard time recovering from their mistakes.  Why, oh, why do we subject students to such pain, other than the native sadism?

Well, in my opinion, our native sadism is reason enough because this sort of pain is good for you, but beyond that, these identities are, in the long-run far more useful than the mensuration formulas we teach.  First, we have to use these identities to prove the mensuration formulas.  There is no royal road to geometry, Mister, if Alexander the Great had to learn it, then you do too.

But more than that, these formulas will be seen again in calculus.  They make certain otherwise impossible problems easy.  In addition,electrical engineers will probably take a course in Theory of Functions of a Complex Variable, and these trigonometric formulas pop up again there.

Indeed, I encountered formulas of trigonometry as deeply in mathematics as algebraic topology, and that is pretty deep indeed.

But the productive pain aspect of it was by far the most important part for me.  School, research, and life itself are places where being able to endure this sort of pain are vital.  In Trigonometry, I learned how to do that and picked up some cool formulas while I was at it.

Shall We Gather at the River

By Bobby Neal Winters

Cowboy looked up from the seat of his skiff at the clear, blue sky and felt the chill wind at it whipped past his face.  It was cold, but no colder than would be expected on a mid-December day.  They’d been no snow yet, but the early morning’s frost, now erased by the sun’s rays, had hinted at things yet to be.

Whenever people asked Cowboy what he was doing out on days like this, he said he was out looking for arrowheads along the creek.  It was a believable story as it was the sort of thing that people did in this part of the country.  And today was a good day for arrowhead hunting in any case. There’d been a long, dry summer which had been relieved recently by some heavy fall rains.  With the grass killed out and the erosion of the rains, there might be some new arrowheads exposed.  

And who knows, he thought, I might even find a few arrowheads.  It was the communing with nature, he liked though.  He loved being in the out of doors with only the sky for a roof.

He pulled his skiff up to the creek bank and carefully stepped out.  It was a beautiful day to be out in the open. The only cloud he saw wasn’t actually a cloud; it was smoke rising from a fire up on the hill a short way ahead.  In other places and times he might’ve thought it was hunters, but here and now Cowboy knew it belonged to Wendell Warthind, who’d been a childhood friend of his father’s.  He had a pretty good idea what Wendell was doing too and he made up his mind to ignore it.  Not ignoring it would make life complicated.  He had every intention of following through with this purposeful ignorance, but events weren’t going to allow it.

Cowboy planted his oar in the mud of the bank and tied his boat to it.  

He’d taken one step, maybe two up the bank when he heard the blast.

The sound could’ve been an explosion.  Cowboy knew what Wendell was doing up there and an explosion was one possible outcome of that.  That wouldn’t happen if Wendel was at his best, but he’d been slipping lately.

But Cowboy’s practiced ear knew differently. While it was technically an explosion, it was one that was more precisely characterized as a shotgun blast.  He pulled is 44 magnum from its holster and began running toward the sound when he heard profanity issuing in Wendell’s voice and several sharper explosions which Cowboy recognized as rifle shots.

He came in sight of an opening in the trees where in he saw Wendall hiding behind a large stump over which he was pointing a thirty-ought-six hunting rifle.  In the middle of the clearing was the campfire whose smoke Cowboy had seen. It was beneath a device that Cowboy recognized as a still.

Wendell hadn’t seen Cowboy yet, so Cowboy slipped behind a tree and looked in the direction that Wendell’s rifle was pointed.  The first thing he saw was a jeep with two flat tires and a couple of bullet holes in the side.  Then he saw a pair of feet beneath the jeep.  He followed those feet up when he saw the top of a blond head.    

When he repositioned himself a little, he saw the whole face and his jaw dropped.  The face belonged to the new preacher over at the Pentecostal Church, the Reverend Mosley.

Mary Beth Mosley.

Cowboy turned his attention back to Wendell  It looked to Cowboy that Wendell had a bead on Mary Beth and that if something weren’t done pretty quick there was going to be a-killing.  It was then that Cowboy pulled out his badge and started talking loud.

“Okay, there, Wendell,” he said with his 44 pointing straight at his father’s old friend.  “This has gone far enough.  Put the gun down.”

Wendell was caught off guard.

“What the hell?” he said as he turned.

“Put that rifle down!”

It came with enough force that Wendell put it down.

Cowboy now turned his attention to the Reverend Mosley.

“Mary Beth,” he said. “Throw that shotgun down.”

Mary Beth stood up from behind the jeep revealing a figure that caused Cowboy to curse it being wasted on a lady preacher.  

She had a double barrel shotgun in her hand.

“Oh, Sheriff,” she said brightly, “I am so glad to see you.”

Cowboy didn’t respond to her brightness.

“Put that shotgun down,” he said.  There wasn’t a hint of a smile.

Her brightness dimmed considerably as she put her shotgun down.

“Okay,” Cowboy said. “You come over here.”

As she got closer to him and farther from the gun, he moved toward Wendell, grabbed his rifle, and tossed it into the woods.  As he did this, he noticed that behind Wendell was a glass three-gallon jug of a clear liquid which he appeared to be protecting like the apple of his eye.

Just at the word was forming in Cowboy’s brain, he heard it coming out of Mary Beth’s mouth.

“Moonshine,” she said. “He’s been making moonshine up here.  I figured that I’d bring my shotgun up here and blow some holes in his still, but he was here.  I was trying to shoot that big jug, but I got him instead.  He had it coming, though, because that’s Satan’s brew.”

It was then that Cowboy noticed that Wendall had been wounded in the leg.

“You all right, Wendell?” he asked.

“I’ve been better,” he said.

Cowboy then went to the jug and smelled it.  

Heaven.

He picked up a tin cup from the ground, tipped a little of the jug’s contents into it, and took a swig.

Rapture.

He looked at the jug, he looked at Wendell’s leg, and he looked at the lady preacher’s jeep.  He finished his cup of whisky with a cough.

“Okay, I need to get you all back to town,” he said.  He reached down and picked up the jug.  With it in one hand and his 44 in the other, he said, “Follow me.”

It only took them a short time to get to the river, but the amount and the viciousness of the fighting made it clear that he wasn’t going to be able to leave them alone together.  Once they arrived, another problem became apparent.

“That is surely Satan’s brew,” the Reverend Mosley said. “We ought to just pour it into the river.”

This caused Cowboy to lick his lips as it was the finest tasting brew he’d had in a long time.  There was no way he could let that happen.  He then looked at the creek and his small boat and a new problem occurred to him.  How was he going to get these two people and the jug of whiskey across the river?  The skiff was only big enough for him and one other thing.

If he took Wendell across first, Mary Beth would dump the whiskey into the creek, and Lord Almighty, that would be a crime.  If he took the whiskey across first and left Mary Beth and Wendell alone together, one might kill the other.  There was only one thing he could do.

“Okay, Mary Beth,” he said, “you come with me.”

He put the jug down, sat in the back of the boat, and invited Mary Beth to sit between his knees which--after looking a little suspicious--she did.

It was a cold day, and it warmed Cowboy up--in more ways than one--to have Mary Beth snuggled between his knees as he paddled across.  Once there, he put her out and made the return trip.  He’d been doing a little thinking on his way across and he’d gotten an idea.  When he got to the other side he had a plan.

“Wendell,” he said, “hand me the jug.”

“Ain’t you afraid she’s going to dump it out?”

“Hand me the jug,” Cowboy repeated.

Wendell gave him the jug, and he went back over.  Once there, he got out, put the jug out of the way on the far bank, and turned to Mary Beth.

“Okay, come back with me,” he said.

“Don’t you trust me with the whiskey?” she asked.  The way she was smiling, she new the answer.

“No,” he said.

She climbed back into the boat and he slipped his knees around her again.  This time there was no protest.  Cowboy rowed back over, and when they arrived at the other side, Cowboy, directed his attention to Wendell again.

“Okay,” he said, “you trade places with her.”

This was done with no more problem than an exchange of hateful stares.

When he rowed Wendell to the other side, it wasn’t nearly as pleasant as it had been with the lady preacher.

“Ain’t you afraid I’m going to run off with the whiskey?” Wendell asked when he got out.  

“Not the way that leg is looking,” Cowboy replied.

He then headed back, fetched Mary Beth, and brought her back, all the while regretting that this would be the last time.  When they arrived at the other side, she didn’t seem to eager to get out either.

Wendell broke the spell, however.

“Where do we go now?” he asked.

“My truck’s up there,” Wendell answered. “I think I’ll take you to the emergency room first and once that’s squared away, I think I’ll take Mary Beth...I mean Reverend Mosley home.”

Cowboy thought he saw Wendell’s eyebrow move ever so slightly.

“Is he going to jail after that?” Mary Beth seemed eager to know.

“You never can tell what’s going to happen next,” he said.  “You never can tell.”

Taking the Measure

By Bobby Neal Winters

Introduction

In mathematics, simple concepts can be quite subtle.  Very early, children take out rulers and begin to measure objects.  A foot ruler will suffice to measure a piece of gum; a yardstick is up to the task of measuring one’s forearm; a tape measure will measure a room. None of this is difficult, though it might become tedious if one worries about accuracy, but that is not my concern.  

Hey, I am a pure mathematician; what can I say?  

Let’s leave the so-called practical concern behind. Consider taking a step from the realm of the concrete to the abstract, from the physical world to the world of pure number.

For example, how would we measure the length of an interval on the number line itself.  This isn’t difficult either.  If we consider the length of the interval of real numbers [1,3], we can recognize immediately that it is two units long.  We figure this out because 3-1=2.  It’s simple.  Indeed, if we want to make a more general formula, the interval of real numbers [a, b] is b-a units long. Only subtraction is involved.

You man have noticed that I am using the notation that [a,b] is the set of all real numbers between a and b, including a and b themselves.  If I write (a,b), this means I am excluding both a and b; (a,b] means I am excluding a; [a, b) means I am excluding b.  And all of these intervals are the same length because they differ only by one or two points.  Recall that, as far back as the time of the Greeks, Euclid was saying “A point is that which has no part,” so one or two (or three or fifty) points removed from an interval have no effect on the total length.  Indeed, one may remove any finite set of points from the interval [1,2] and the result will still be two units long.  We aren’t sweating the small stuff.

The sharp reader--and I know you are out there--may have noticed during the course of that last paragraph that I slipped from the concept of the length of an interval to the length of a set. That’s one small step for a man, but one giant leap for Mankind; or Womankind; on Man-unkind if you are e.e. cummings.

In any case, that one small slight of hand opens a whole different kettle of fish.  In order for the general reader to appreciate the problem, I will now spend a little while talking about a simple set of numbers that many otherwise gentle people hate with a fury: fractions!

 

Being Rational

Fractions, I will be talking about them, are better known as rational numbers.  As I am no longer talking about this on a street corner or in a bar--pssst,wanna take a look at my square root?-- I’d best call things by their scientific names.  Rational numbers are numbers of the form m/n where m and n are both integers but n is not zero.  (Remember, dividing by zero makes green hair grow on your palm. Er...never mind.)  I mentioned earlier that many otherwise gentle people hate rational numbers: mathematicians love them.  Mathematicians are rather like cops, soldiers, morticians, grave robbers, or other people whose jobs force them to see things other people don’t have to, and they have a different standard of nastiness.  Compared to  some of the critters we’ve seen, rational numbers are as tame as puppies.

One thing that most people don’t know is that rational numbers are literally everywhere on the real number line.  Between any two real numbers there is a rational number.  This is easy to see; those of you who aren’t up to a dose of algebra can skip the next paragraph.

Take any two real numbers a less than b.  We know that b-a is positive.  One can choose a positive integer n to be large enough so that n times b-a is greater than one.  This means that there is an integer m that lies between n times a and n times be.  It follows that m/n is between a and b.

Mathematicians have a name for this property of the rational numbers.  We say that the rational numbers are dense in the real numbers.  Mathematicians are also frequently referred to as being dense, usually by their spouses, but that is a matter for another day.

 

The Real Thing

I’ve been referring to the real numbers as well as the rational numbers, and the place that they live together in peace and harmony as the number line.  When I was young and full the brashness rightly common to all newly minted PhDs, I understood all this stuff.  As I’ve had a been of that brashness scuffed off me, I’ve lost much of that understanding.  I try now to approach numbers with a child-like humility.  All of this to say that I am now going to explain some things that bright people like yourselves already know.

All rational numbers are real numbers, but not all real numbers are rational: pi is irrational; the square root of two is irrational.  Those are two examples, but there are lots of others.  Most numbers are irrational. When I say that, I could mean any number of different things.  One thing would refer to the concept of number.  The set of rational numbers is countably infinite, but, by way of contrast, the set of real numbers is uncountably infinite.  I’ve talked a bit about this in other essays.  Another thing that I might mean is that the real numbers are infinitely long and the rational numbers have a length of zero.

In case you are surprised that the rational numbers have a length of zero, you are not alone.  Indeed, one of the early methods of attempting to measure sets called “content” measured the rational numbers as being infinitely long as well.  There were problems with content, however.

 

Piecing It Together

The problem with content can be easily explained.  The content of the rational numbers between zero and one is one; the content of the irrational numbers between zero and one is one.  Now, when I take something that is a foot long and stick it together with something else that is a foot long, I don’t expect the result to still be a foot long; I expect it to be two feet long.  It is at this point that we can start making excuse for poor little content, but that would take me too far afield.  Suffice it to say that content just doesn’t measure up.  (If you missed the pun in the last sentence, I urge you to go back and re-read it.  I am very proud of it.)

We want the sum of the length of disjoint sets to be equal to the length of all of those sets put together, and we wan that to work even when we join together a countably infinite number of sets. A fellow named Lebesgue figured out how to do this and it works.  He cleared up a number of problems that we were having with some very technical mathematics at the same time.  We mathematicians love those among us who clear up technical problems.  We tend to name things after them and teach courses about those things.

I teach a course about what Lebesgue did.  It takes me quite a while to get all of the technical details lined up for the graduate students who take it.  I could dump that all on you right now, but you might whimper; my grad students do.  

Rather than may you whimper, let me show you something cool that was opened up by our old friend Lebesgue.

 

The Un-Measurable

Once upon a time, there was an Italian mathematician by the name of Giuseppe Vitali.  (Let’s just call him Joe, okay?) Joe created a set that not even Lebesgue could measure.  I will show you the set before I let you go.  Seriously, if you haven’t noticed, I’ve trapped you in your seat now.  You are mine! You are mine I say! Bwahahaha!

Er.  Sorry.

Suppose that you could create a set V with the following properties:

1. V is contained in the interval [0,1);
2. There are sets V1, V2, V3, … each geometrically congruent to V such that
1. they are disjoint from each other;
2. there union contains [0,1);
3. there union is contained in [-1,2).

One might innocently look at those conditions and not think much of it.  It all seems so utterly reasonable.  Well, that just goes to show what you know.

Suppose we call the union of all of these sets U.  As union begins with U this sort of works.  On one hand, since U is contained in [-1, 2) and that interval is no longer than three units long, we know that U is no longer than three units long.  On the other hand, since U contains [0,1) and that interval has a length of one unit, we know that U is at least one unit long.

So far, so good.  It is about to get messy.  

Now if things add up the way they are supposed to, the length of U should be the sum of the lengths of V1, V2, V3, … , right?  As each of these sets is congruent to V and they are all disjoint, this should be easy.  We are just adding the length of V to itself an infinite number of times, right?

Again, the alert reader’s eyebrows may just have arched in a Vulcan-like fashion. This is problem that even those pesky Greeks knew about.  If you add up the same number (non-negative) an infinite number of times, the sum is either zero or infinity.  It’s zero if the number you are adding together is zero, and it’s infinity if is positive.  

So if we there is a set V that has the properties of the of one described, either one is less than zero or three is greater than infinity, or, at least, so it seems.

The faint hearted would stop at this point, but not our Joe.

Joe created V just as we describe above, and he did it in stages.  

Joe first broke [0,1) into some disjoint sets.  The first of these was the set of all rational numbers in that interval.  The rest he obtained by taking copies of the rational by sliding them and rotating them through [0,1).

What’s that? A hand went up in the back.  What do I mean by sliding and rotating?

By sliding I mean that you act as if the rationals are marked on a clear ruler the is laying over [0,1) and then sliding that along to cover other numbers.  By rotating, I mean that when we push the numbers past 1, act as if the interval in bent in a circle back on itself and those numbers pop up past 0.  

“It’s all on a circle, Grasshopper.”

Anyway, once you’ve broken the interval [0,1) into pairwise disjoint sets in that way, you choose one number from each of those sets and form V from those points.

The studious drudges who occupy the ranks of the mathematical world can verify that V satisfies the demands that we’ve put on it.  

We are left with a crisis.

 

Resolving the Crisis

I’d said above that we were left with the choice that either one is less than zero or three is greater than infinity.  I left out an alternative.  That is that there are just some sets that can’t be measured. The set V is such a set.

In solving problems, mathematicians often find more than they bargained for.  This is such a case. Above, quite innocently, I told you to form V by just picking a point from each one of the disjoint sets described.  That particular action is now referred to as the Axiom of Choice.  It is a method of creating sets that people hadn’t given too much thought to up until that point.

They started thinking about it.

I may as well, but I will be like Scarlett O’Hara and worry about it another day.

Teaching, Writing, and Mathematics

By Bobby Neal Winters

I have been thinking recently about the writing of mathematics.  This is rendered to being a highly academic exercise as I have not produced any original mathematics to write about for quite some time.  I don’t bring very much credibility with me to this endeavor because during the brief interval I was producing original mathematics I cannot say that I was a very good expositor of it.  

What has happened in to bring my thoughts to the writing of mathematics?  One thing is that I have begun writing myself.  I began writing a weekly column for the local news paper about 10 years ago.  Figure 52 columns a year--I never miss a week--and seven hundred words a column--I usually overrun that barrier--and this comes to about 364,000 words.  I write things besides my column so I will say that I’ve produced about half a million words in the last ten years.  If repetition is the mother of learning--and I believe it is--then I may have learned something about how to write.

In addition to this, it has been my pleasure to teach out of a pair of very well-written books: A Radical Approach to Real Analysis and A Radical Approach to Lebesgue’s Theory of Integration, both written by David Bressoud.  These books are, as I said, well-written, but I think more important to their effect on me is they are written as historical approaches to their particular subjects.  

The historical approach has been very enlightening to me.  Mathematics is a land populated by optimizers and those seeking efficiency through brevity.  A result is discovered and proven with great difficulty.  Then time is spent in organization working out theory wherein the proof of the result seems not only easy but inevitable.  The historical approach allowed me to appreciate that what has been rendered as clear as glass to students of the subject today was once a riddle in a mirror.

As a teacher, it reawakened the excitement of the subject in me.

But in presenting this material to my classes, I’ve reproduced the older proofs of various results that have been included and in doing so I’ve noted a difference in style with much of modern mathematics.  There is a tendency in modern mathematics to drift toward the abstract.  This is a reasonable tendency as the abstract proofs tend to be cleaner and tend to give broader results, i.e. if I prove something about metric spaces, then I prove it about the real numbers, the complex plane, and spaces of functions at the same time.

That is something of an illusion, however.  The abstract structures are out-growths of specific examples.  The examples were originally things that were interesting in themselves and drove the construction of theory to discuss them of abstract structures to more easily prove results about them.

It is somewhat ironic that my experience as a mathematician helped me to be a better writer of things besides mathematics.  As a topologist who studied 3-manifolds, I took trips in my head to places no one had been before and then attempted to explain them to those who’d remained at home.  I learned the art of description as I described these strange places. In the construction of proofs order is important.  In good writing, some things must be explained before others can be understood.

As a writer, it seems to me that in well-written mathematics, well-constructed examples serve the same purpose that metaphors do in good writing.  Much writing seeks to create in you a picture that I have seen in me.  This is true whether I am writing about house cats, building a computer, or mathematics.  If I am to be successful, I must reach you by beginning with a thing you already understand and build from it.  Our shared experiences make communication possible.  Mathematically, the creation of a well-chosen example gives writer and reader a common, shared experience from which the writer may then deviate in order to build.

This technique is especially evident in the proofs in analysis that begin with the proof of a simple, very special case and proceed through stages of increased generality until the most general case is proved.  At its best, one can see the whole of the problem is in that very special case and the succeeding generalizations are simply minor modifications of the original proof.

There have been times when I’ve complained that too much of the time we who teach mathematics treat it as a murder mystery.  We withhold or downplay certain very important details.  We keep things to ourselves as our own little secrets.

I think I know why we do this.  We are trying to teach our students how to think for themselves.  There is a reluctance to “lay out everything plain” because it will deny the students the joy of the gestalt that we ourselves experienced, the joy that led us to become mathematicians.

I must say that there is much virtue to this point of view.  The problem is that it is so easy to do very, very badly.  There is also a tendency--among some in the profession--to use this as a technique to build up their own egos at the expense of their students’. This is far from universal, however.  Much more of a problem is judging difficulty.  What is easy to a professional might be insurmountable to a student.  The teacher’s job is to lay out bread crumbs to tempt the student to the point of gestalt but not so many as to rob the student of the joy of that gestalt. The fat cat will catch no rats.  Not that a cat eats bread crumbs, but sometimes you have to mix your metaphors.

In our teaching, in our writing, we must free ourselves to ape literature.  We we must foreshadow the great mathematical truths to come with smaller, more easily digestible truths.  When the student finally comes to the climax, he must be so prepared that the final step to him is nature, the final joy is true, but he should then be able to reexamine his steps and realize his arrival at this particular point was no accident.

I mentioned at the beginning of this article that when I was producing original mathematics I wasn’t a particularly good expositor of it.  There are those who will read that sentence and recognize me as a master of understatement.  While a careful study of my articles would undoubtedly produce additions to this list, I believe I had two major problems: impatience and an over-reliance on notation.

I believe we can all agree that taking one’s time to do a good job is a virtue.  A lack of patience can cause a lack of proper care.

Creating notation is a way around the difficulties we sometimes encounter in natural language.  Creating a well-chosen metaphor, a well-created example is another.  The example has the advantage that someone might name it after you one of these days.

If I were suddenly given my mathematical life to live over, I hope that I would choose to grasp onto mathematical exposition as the true art that it can be.  To first live the mathematical adventure and then tell the tale, and to understand the telling of the tale is as at least as important as the adventure because its purpose is to convince others to have adventures themselves.

(Bobby Winters is Assistant Dean of the College of Arts and Sciences and Professor of Mathematics at Pittsburg State University. He now holds the title of University Professor.)